Performance analysis of an adaptive feedback active noise control based earmuffs system

Performance analysis of an adaptive feedback active noise control based earmuffs system

Applied Acoustics 96 (2015) 53–60 Contents lists available at ScienceDirect Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust Pe...

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Applied Acoustics 96 (2015) 53–60

Contents lists available at ScienceDirect

Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust

Performance analysis of an adaptive feedback active noise control based earmuffs system Seong-Pil Moon, Jeong Woo Lee ⇑, Tae-Gyu Chang * School of Electrical Engineering, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul, South Korea

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 13 December 2013 Received in revised form 26 February 2015 Accepted 11 March 2015 Available online 30 March 2015

This paper analyzes the performance of the adaptive feedback active noise control (FBANC) based earmuffs system. The system’s noise reduction performance is obtained in a closed-form equation with respect to two top-level design constraining parameters, i.e., secondary path delay and noise bandwidth. To derive the equation, we utilize a simplified and equivalent linear prediction ANC model and parameterize noise bandwidth using an autoregressive model. The derived equation is validated by simulations and experimental tests performed for the ANC earmuff system. The high degrees of noise reduction, which are 14.9 dB for the airplane noise and 20.4 dB for the house heater noise, obtained from the experimental tests confirm the feasibility of the ANC earmuffs application. Ó 2015 Elsevier Ltd. All rights reserved.

Keywords: Active noise control Earmuffs Linear prediction Secondary path delay Noise bandwidth Noise reduction

1. Introduction Active noise control (ANC) earmuffs are intended to suppress the external noises whose dominant components are at relatively low frequency band, such as airplane engine noise, rotating machine noises in plants, house heater noise, etc. [1–4]. Passive earmuffs alone are ineffective to suppress the noise components at the frequency range below several hundred Hertz and tend to be very expensive and bulky [5–7]. Many commercial ANC headsets applications are developed based on the feedforward ANC scheme. However, the feedforward ANC systems for headsets have significant stability and performance deficiencies caused by nonstationary reference inputs, measurement noise, and acoustic feedback [1,6,8–10]. Unlike the feedforward ANC, the adaptive feedback active noise control(FBANC) provides a more accurate noise cancelation since the microphone is placed inside the ear-cup of the headset [10]. The functional structure of the FBANC earmuffs system is illustrated in Fig. 1. Most of previous works performed in regards to the feedback ANC architecture are the transfer-function-model-based approaches to address the stability, sensitivity and robustness issues of the system [11–14]. In general, the model-based ⇑ Corresponding authors. E-mail addresses: [email protected] (J.W. Lee), [email protected] (T.-G. Chang).

(S.-P.

http://dx.doi.org/10.1016/j.apacoust.2015.03.006 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.

Moon),

[email protected]

approaches do not directly provide explicit and geometric interpretations in the domain of physical parameters, such as geometric parameters of noise canceling space, electro-acoustic coupling delay, noise characteristics parameters, etc. On the other hand, most of previous works addressing the explicit effects of physical parameters rely on, instead of an analytic approach, direct simulations to show the effects in the noise reduction of the ANC system [13–15]. In this paper, two key physical parameters of FBANC, i.e., secondary path delay and noise bandwidth, are selected and their combined effects are analytically investigated to obtain a closedform noise reduction equation. The secondary path is named for the electro-acoustic coupling between the speaker system and the sensing microphone system. The delay in the secondary path significantly degrades the noise reduction performance of the ANC system. Therefore there exists a tradeoff between the noise reduction performance and the size of spatial coverage range of noise canceling [13,16]. The performance degradation becomes significantly aggravated when the delay becomes longer than the coherence-time of noise correlation. The coherence time is inversely proportional to the noise bandwidth. Therefore, it is very meaningful to investigate the combined effect of secondary path delay and noise bandwidth on the performance degradation of the FBANC earmuffs system. In order to demonstrate the exclusive effects of aforementioned two parameters, a simplified and equivalent D-step linear predictor model of the FBANC system is used for the performance analysis of

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A typical structure of the adaptive feedback ANC system is shown in Fig. 2. The FBANC system suppresses the primary noise dðnÞ by generating the anti-noise yðnÞ with the synthesized reference signal xðnÞ. The secondary path SðzÞ represents the electro-acoustic coupling path from the loudspeaker to the error microphone. ^ SðzÞ is the internal estimated model of SðzÞ, which is ~ used for the estimation of primary noise dðnÞ. The simplified D-step linear predictor model of the FBANC is obtained by assuming SðzÞ ¼ zD and ^ SðzÞ ¼ SðzÞ. The simplified

Fig. 1. Functional structure of an ANC earmuffs system.

the FBANC. The explicit inclusion of the delay as the structural parameter of the predictor model, i.e., the model’s D-step prediction distance, enables a straightforward derivation of the exclusive effect of the delay on the performance degradation. The noise bandwidth is parameterized using the second order autoregressive model and applied to the linear predictor model for deriving a closed-form noise reduction equation. This approach provides a useful result in the sense that two key design and application constraining parameters, i.e., secondary path delay and noise bandwidth, are explicitly included in the closed-form equation showing their combined effects. Computer simulations and experimental tests are performed to verify the analytic derivation of the noise reduction performance regarding the FBANC earmuffs system. Recorded airplane noise and house heater noise are also used as the primary noise in the tests to show the applicability of the analysis.

model is shown in Fig. 3, where the order of zD and WðzÞ can be changed under the assumption of quasi-stationarity of the primary noise. Such structural simplicity enables the straightforward derivation of the exclusive effect of the delay and the noise statistics. Even though the structure of the linear predictor model seems over-simplified, the utilization of the simplified model is justifiable considering that the disturbance of model parameters should be assumed to be controlled by the feedback mechanism and most of all, the use of a nominal plant model is reasonable for the purpose of performance bound analysis [6]. Since the model has an entirely feedforward structure, the theoretical maximum noise reduction (NR) bound can be readily derived in the sense of Wiener as

NRmax

h i h i E jdðnÞj2 E jdðnÞj2 i¼ h i ¼ h E jemin ðnÞj2 E jdðnÞ  wHo xðnÞj2 ¼

r2d

ð1Þ

r  rðDÞH R1 rðDÞ 2 d

where wo ¼ R1 rðDÞ: the optimum filter coefficient vector,

emin ðnÞ : the minimum prediction error; xðnÞ ¼ ½dðn  DÞ dðn  D  1Þ



T

dðn  D  M þ 1Þ ;

E½ : expectation operator; E½dðnÞ ¼ 0;

r2d ¼ E½dðnÞd ðnÞ;

  R ¼ E xðnÞxH ðnÞ ;

2. Derivation of noise reduction performance of the adaptive feedback ANC based earmuffs system

rðDÞ ¼ E½xðnÞd ðnÞ ¼ ½rðDÞ rðD  1Þ

This section is devoted to a derivation of noise reduction performance of the FBANC earmuffs system. In the derivation, the FBANC’s equivalent linear predictor model and the autoregressive model based primary noises are used. In the following Sections 2.1 and 2.2, the overall procedures of the simplified derivation method and its derived result are presented respectively. In the Section 2.3, the overall effects of secondary path delay and noise bandwidth are shown with an exemplary illustration of performance curves.

The autocorrelation vector rðDÞ in (1) consists of the autocorrelations of the primary noise with the time lag from D to ðD þ M  1Þ, which is the D-step predictor’s filtering window duration. Our motivation of investigating the combined effect of secondary path delay and noise bandwidth is based on (1), where the noise reduction determining factor rðDÞ depends on the relative length between the secondary path delay D and the effective time lag width of autocorrelation which is the inverse of the noise bandwidth. Specifically, the noise reduction performance





rðD  M þ 1ÞT :

2.1. Linear predictor model based performance derivation This section describes the details of the linear predictor model based performance derivation. In the derivation, the feedback ANC is simplified to an equivalent D-step linear predictor with the assumption that the secondary path is a pure D-step delay and its estimation is errorless [1,6,9]. By exploiting the prediction error of the D-step linear predictor, the noise reduction performance of the ANC can be derived to examine the combined effect of the secondary path delay and the coherence time of the noise correlation.

Fig. 2. Block diagram of a typical adaptive feedback ANC system.

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prediction error of the D-step linear predictor. The minimum mean-squared prediction error can be obtained by calculating the amount of the innovations that occur during the latest D-step interval. Such innovations cannot be utilized in the D-step linear predictor. The minimum mean squared prediction error is expressed by ^o ðnjdnD Þ as using its D-step optimum predictor d Fig. 3. Equivalent D-step linear predictor model of the feedback ANC system.

significantly drops when the secondary path delay D becomes relatively long such that the filtering window moves out from the effective time lag width of autocorrelation. The secondary path delay should be short enough to be within the time range where the autocorrelation function remains high, equivalently where the primary noise is highly correlated in order to achieve a high degree of noise reduction. 2.2. Derivation of noise reduction performance of FBANC This section presents the derivation procedures of a closed-form noise reduction equation in terms of secondary path delay and noise bandwidth. The noise bandwidth is parameterized by adopting the autoregressive (AR) model and applying it to the linear prediction model. This methodology of noise bandwidth parameterization enables the analytic derivation of the combined effects. This AR noise model based derivation can also be understood as one way of characterizing the noise reduction Eq. (1) in the physical parameter domain, i.e., the electro-acoustic coupling delay and the noise bandwidth. By including the pole location parameters in the AR model coefficients, the physical parameters for characteristics of noise, i.e., center frequency, bandwidth, and Q-factor, can be explicitly reflected. As shown in Fig. 4, the pole location of the second order AR model directly indicates the 3 dB bandwidth of the noise B3dB , where B3dB  2ð1  lÞ in radians per sample under their narrowband assumption, i.e., 1  l  1, and l is the pole radius in the range. The second order AR noise having two poles at z1 ¼ l  ejx0 and z2 ¼ l  ejx0 is defined as [17], 2

dðnÞ ¼ 2l cosðx0 Þ  dðn  1Þ  l  dðn  2Þ þ iðnÞ;

ð2Þ

where iðnÞ is a white Gaussian noise with variance r2w . The maximum noise reduction for the second order AR model h i can be derived by rewriting E jemin ðnÞj2 in the denominator of

h i E jemin ðnÞj2 ¼ r2d  rðDÞH R1 rðDÞ ¼ r2d  rðDÞH wo h i H ¼ r2d  E dðnÞd ðn  DÞwo h i ^ ðnjdnD Þ ¼ r2d  E dðnÞd o  2  ^  ¼ r2d  E d o ðnjdnD Þ :

ð3Þ

^o ðnjdnD Þ can be expressed by the best The optimum predictor d linear combination of innovations contained in the D-step past value of the dðnÞ as [18]

^o ðnjdnD Þ ¼ d

M1 X

wo ðkÞdðn  k  DÞ ¼

k¼0

1 X hðkÞiðn  kÞ;

ð4Þ

k¼D

where hðnÞ is the impulse response of the second order AR generation filter. From (2), the transfer function of the second order IIR filter HðzÞ for generation of the second order AR model [17] is expressed as

HðzÞ ¼

1 2

1  2l cosðx0 Þz1 þ l z2

:

By taking the inverse z-transform of HðzÞ, the impulse response hðnÞ is obtained as n

hðnÞ ¼

l sin xo ðn þ 1Þ sin xo

for n ¼ 0; 1; 2; . . .

ð5Þ

The minimum mean squared prediction error is derived by plugging (5) into (4), then applying the result to (3). The detailed derivation of the minimum mean-squared prediction error h i E jemin ðnÞj2 is given in Appendix A. The result is expressed as h i 2D 2 E jemin ðnÞj ¼ r2d  r2d l " # 2 2 sinð2xo Þsinð2x0 DÞ þ 2 sin ðx0 DÞðcos2x0  l Þ  1þ : ð6Þ 2 2 2 2 sin x0  ð1 þ l Þ=ð1  l Þ

The maximum noise reduction is obtained as

(1) using the AR model’s theoretical minimum mean-squared

NRAR: max

h i E jdðnÞj2 i ¼ h E jemin ðnÞj2 ¼

2 2D dl

r2d  r

¼ 1l

2D

h

h

r2d 2

2

xo DÞþ2 sin ðxo DÞðcosð2xo Þl 1 þ sinð2xo Þ sinð2 2 sin2 ðx Þð1þl2 Þ=ð1l2 Þ

Þ

i

o

1 2

xo DÞþ2 sin ðxo DÞðcosð2xo Þl 1 þ sinð2xo Þ sinð2 2 sin2 ðx Þð1þl2 Þ=ð1l2 Þ

2

i;

Þ

ð7Þ

o

which provides the upper bound on the noise reduction in terms of the secondary path delay D and the bandwidth of the second order autoregressive noise. Here, the 3 dB bandwidth of noise is directly indicated by the distance from the poles to the unit circle in the zplane, i.e., 1  l. 2.3. An example of the feedback ANC performance curves Fig. 4. Parameterization of 3 dB noise bandwidth using the pole location parameter of the second order autoregressive model.

In this section, the overall effect of the secondary path delay and noise bandwidth is shown by an exemplary illustration of

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performance curves obtained from (7). These performance curves can be used as a top-level design guide to determine the upper bound on the electro-acoustic coupling delay and the target range of noise bandwidth for suppression. The curves indicating the achievable degree of noise reduction in dB are illustrated in Fig. 5. The range of noise bandwidth varies from 0.1 Hz to 100 Hz, and the range of secondary path delay increases from 0.125 ms to 20 ms. The center frequency is set at 250 Hz to reserve the sufficient separation between two peaks of noise. It is confirmed in Fig. 5 that as the target degree of noise reduction increases, secondary path delay and noise bandwidth should decrease. As an example, for three different noises having bandwidths of 1 Hz, 10 Hz and 20 Hz, the allowed ranges of secondary path delay can be found in Fig. 5. Three points A, B and C on the 10 dB target line indicate that the allowed maximum delays are 17 ms, 1.32 ms, and 0.87 ms, respectively. In order to attain 20 dB of noise reduction, the allowed maximum delays are reduced to 1.25 ms, 0.42 ms, and 0.32 ms, respectively, as indicated by A0 ; B0 and C0 on the 20 dB target line. For a given target degree of noise reduction and noise bandwidth, the maximum allowed distance of the electro-acoustic coupling can be found from the tradeoff relationship shown in Fig. 5. This is a good example of top-level design guide showing the relationship among the target level of noise reduction, noise bandwidth, and the electro-acoustic coupling distance.

3. Computer simulations In order to verify the theoretical analysis derived in the previous sections, the FBANC earmuffs system was simulated using the second order AR noises and the recorded airplane and house heater noises as the primary noises. The earmuffs simulation system is implemented using the filtered-x LMS(FxLMS) based FBANC algorithm, for structural conformity with the experimental setup in Section 4, as shown in Fig. 6 [1,9]. Here, the secondary path SðzÞ and its estimated model ^ SðzÞ are set as a pure delay model, zD to maintain the equivalence of the FBANC based simulation model with the simplified feedforward model used for the theoretical analysis in Section 2. The reference signal xðnÞ is internally synthesized using the feedback mechanism where the residual signal eðnÞ and the filtered

Fig. 6. The block diagram of the filtered-x LMS algorithms based Feedback ANC system used in the computer simulation.

anti-noise signal y^0 ðnÞ are feedback and added. The normalized least mean square (nLMS) algorithm [19] is used to update the adaptive filter WðzÞ running at 8 kHz. The filter length is set to include the secondary path delay. The agreements between the analytic results of (7) and the simulated results are well demonstrated in Fig. 7. The second order AR model was used to generate the primary noises having bandwidths of 1 Hz, 10 Hz and 100 Hz. The secondary path delay was incremented from 0.25 ms to 6.25 ms. In the analytic results (7), the excessive error of the LMS algorithm [19] was reflected for the comparison with the simulation results. The effectiveness of the FBANC earmuffs system is well illustrated with the simulation results obtained for the recorded airplane noise and house heater noise that are considered as typical targets of ANC [20,21], as shown in Figs. 8 and 9, respectively. Isolation effect of the passive earmuffs is reflected by applying a first-order Butterworth lowpass filter having the cut-off frequency at 200 Hz as exemplified in commercially available earmuffs. The filtered airplane noise and house heater noise are applied to the FBANC algorithm as the primary noise dðnÞ. Power spectra of the residual noise eðnÞ for the airplane noise and the house heater noise are shown in Figs. 8 and 9, respectively. The power spectra of the original noises and those of the initially isolated noises by the passive earmuffs are also shown together with those of the residual noise. It is shown in the Figs. 8 and 9 that more than 20 dB noise reductions are achieved below several hundred Hertz with the use of a headphone type earmuffs system having the secondary path delay of 0.25 ms.

Noise reduction (dB) 0.1

40

1

0.

20 10

1

2.0

6

15

1 3

1.0

B

A’

C’

0.25 0.125 0.1

20

25

30

6

10 15

B’

3

6

C

10 15 20

10

10

20

25 20 15 10 5

15 20

25

1

30

Noise reduction (dB)

3 6

0.1

3

Simulation BW=1Hz Analysis BW=1Hz Simulation BW=10Hz Analysis BW=10Hz Simulation BW=100Hz Analysis BW=100Hz

35

1

1

5.0

25

Secondary path delay (msec)

1

3

A

10

15

20

10.0

6

10

15

20.0

100

Bandwidth (Hz) Fig. 5. Illustration of noise reduction bounds vs. noise bandwidth and secondary path delay.

0 −5

0

1

2

3

4

5

6

Secondary path delay (msec) Fig. 7. Comparison of the noise reductions of the FBANC (simulation vs. analytic derivation).

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Original airplane noise

60

Noise after passive earmuffs

Power Spectrum (dB)

50

Table 1 Comparison of noise reductions for the recorded airplane and house heater noises (the analytic results vs. the simulation results).

Residual noise after FBANC active earmuffs

Delay

40

0.25 ms

30 20 10 0

0.5 ms

1 ms

Noise reduction of airplane interior noise Analytic results (AR noise) 22.84 dB Simulation 23.08 dB

15.53 dB 15.31 dB

8.33 dB 8.03 dB

Noise reduction of house heater noise Analytic results (AR noise) 31.10 dB Simulation 27.47 dB

23.52 dB 20.34 dB

15.88 dB 13.69 dB

−10 −20 −30

50

100

200

400

800

Frequency (Hz) Fig. 8. The simulation results for the recorded airplane noise: power spectra of the airplane interior noise (original noise), the passive earmuffs-applied noise (primary noise dðnÞ), and the active earmuffs-applied residual noise (eðnÞ).

Simulations with the recorded airplane noise and house heater noise are expanded to find the noise reduction performance for three different coupling dimensions of the ANC applications. Here, three different values of secondary path delay, i.e., 0.25 ms, 0.5 ms and 1 ms, are associated with the FBANC earmuffs having different sizes of zone of quiet (ZOQ) and different latencies of the analog to digital conversion process [16,22,23]. The feasibility of the FBANC based noise reduction technique is confirmed from the simulation results shown in Table 1. The noise reduction of several tens of decibel for the 0.25 ms and 0.5 ms secondary path delays and above 8.03 decibel even for the 1 ms delay can be considered significant degrees. The recorded airplane noise and the house heater noise are fitted to AR models and the AR models are applied to (7) to obtain the analytic results and to compare them with the simulation results as included in Table 1. The airplane interior noise is fitted to have the center frequency at 129 Hz and the 3 dB bandwidth of 66.24 Hz. The house heater noise is fitted to have the center frequency at 146 Hz and the 3 dB bandwidth of 7.64 Hz.

4. Experiments Experimental tests of the FBANC earmuffs system are conducted to verify the analytically derived noise reduction Eq. (7) and to show the feasibility of its practical application. The same noises as in the simulations of the previous section, i.e., the second order AR modeling noises and the recorded airplane and heater noises, are acoustically generated with a loud speaker to be used in the experimental tests. The overall experimental setup was implemented with a dummy head mounted FBANC earmuffs system, a primary noise generation speaker, and a residual noise recorder with an independent computer as shown in the Figs. 10 and 11. Here, the dummy head mounted earmuffs system is constructed with referencequality headphones, and a measurement microphone. The FBANC algorithm is implemented on a floating-point TMS320C6x DSP system. An external AD/DA conversion daughter board is also used with the DSP system to provide the low-latency AD/DA converting. The AD/DA conversion system is set to run at 48 kHz. Its corresponding total latency of the AD/DA conversion is measured as 0.20 ms. The FBANC algorithm is operated at

Original house heater noise Noise after passive earmuffs Residual noise after FBANC active earmuffs

60 50

Power Spectrum (dB)

As confirmed from the comparison with the airplane noise showing difference only less than 0.3 dB, the closed-form noise reduction Eq. (7) derived in Section 2 is valid. In the house heater noise case, the difference between the analytic results and the simulation results is relatively larger than that of the airplane noise case, showing a difference of up to 3.63 dB. This greater difference is caused by the poor match of the second order AR model with house heater noise, which contains a significant amount of spread peaks as shown in Fig. 9.

40 30 20 10 0 −10 −20 −30

50

100

200

400

800

Frequency (Hz) Fig. 9. The simulation results for the recorded house heater noise: power spectra of the airplane interior noise (original noise), the passive earmuffs-applied noise (primary noise dðnÞ), and the active earmuffs-applied residual noise (eðnÞ).

Fig. 10. A picture of the experimental setup of the FBANC based earmuffs system.

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Fig. 11. A functional block diagram of the implemented FBANC based earmuffs and its performance measurement system.

8 kHz by downsampling the AD converted signal at the rate of 6 to 1. The length of the ANC adaptive filter is set to 64 taps. The secondary path delay, by adjusting the buffer delay of D/A converter, is set to 0.5 ms and 1.0 ms. The secondary paths having the two different delays are modeled using 32-tap FIR filters to represent the secondary path estimation model ^ SðzÞ. To set the experimental environment close to the simplified model environment used in the analysis, where the secondary path is a pure time delay, a 31-band analog equalizer is also installed for fine tuning of the headphone speaker’s frequency response. In Fig. 12, comparisons of the magnitude responses of the headphones with and without the 31-band equalizer are shown. As shown in Fig. 13 and Table 2, the experimental results performed with the second order AR model noises having the bandwidth from 1 Hz to 100 Hz show good agreements, i.e., less than 1.5 dB differences, with the analytic results of (7). The experimentally measured noise reduction performance for the real airplane and house heater noises are summarized in Table 3. The results are compared with those of the simulation and the analytically derived equation. For 0.5 ms of secondary path delay, high degrees of noise reduction, i.e., 14.9 dB for airplane noise and 20.4 dB for house heater noise, are achieved to show

Fig. 13. Comparison of the noise reductions for the second order AR noises (the experimental results vs. the simulation results vs. the analytic results).

Table 2 The experimental results for the second order AR noises (compared with the analytic results and the simulation results). Secondary path delay

Fig. 12. Comparison of the magnitude responses of the headphones with and without a 31-band equalizer.

3 dB bandwidth 1 Hz

10 Hz

100 Hz

0.5 ms Analysis Simulation Experiment

28.06 dB 26.60 dB 26.65 dB

18.15 dB 17.92 dB 16.63 dB

8.91 dB 8.83 dB 8.31 dB

1.0 ms Analysis Simulation Experiment

21.50 dB 21.00 dB 21.02 dB

11.67 dB 11.45 dB 11.08 dB

3.23 dB 3.07 dB 2.72 dB

its feasibility of practical applications. Even for the longer secondary path delay of 1.0 ms, significant degrees of noise reductions, i.e., 8.1 dB for airplane noise and 13.4 dB for house heater noise, are also achieved. The analytically derived equation of noise reduction performance, i.e., Eq. (7), is validated from the comparison results shown for the airplane noise in Table 3. The experimental results for the airplane noise, which fits well to the second order AR model, show the difference only less than 0.6 dB when compared to those of analysis and simulation. The validity of the derived equation and

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S.-P. Moon et al. / Applied Acoustics 96 (2015) 53–60 Table 3 The experimental results for the recorded airplane and house heater noises (compared with the analytic results and the simulation results). Secondary path delay

Airplane noise

House heater noise

0.5 ms Analysis Simulation Experiment Experiment (w/o equalizer)

15.53 dB 15.31 dB 14.91 dB 14.66 dB

23.52 dB 20.34 dB 20.42 dB 19.41 dB

1.0 ms Analysis Simulation Experiment Experiment (w/o equalizer)

8.33 dB 8.03 dB 8.08 dB 8.00 dB

15.88 dB 13.69 dB 13.39 dB 12.95 dB

domain parameters, i.e., the secondary path delay and the noise bandwidth, are explicitly included in the closed-form equation. The derived equation provides a useful design guide by showing the relationship among top-level design parameters of the FBANC earmuffs system. The derived equation of noise reduction for the FBANC earmuffs was verified through computer simulations and experimental tests performed with the recorded airplane and house heater noises. The high degrees of noise reduction obtained from the tests with the recorded noises, i.e., 14.9 dB for the airplane noise and 20.4 dB for the house heater noise, also support the feasibility of practical application of the FBANC earmuffs. Acknowledgements

simulation is also confirmed from the experimentally obtained data for the house heater noise. As shown in Table 3, the experimentally measured noise reduction for the house heater noise shows the differences of only 0.1 dB and 0.3 dB for the 0.5 ms and 1.0 ms secondary path delays, respectively, when compared with those of simulations. It is also worth to mention that the modeling mismatch of house heater noise, which contains relatively large amount of harmonic components, causes the performance deviations of the experimental measurements up to 3.10 dB, and 2.49 dB for the 0.5 ms and 1.0 ms secondary path delays, respectively, when compared to the results of Eq. (7). Experiments are also conducted without using the 31-band equalizer, which was installed to make the headphones frequency response flat. As shown in Table 3, the noise reduction performance is slightly degraded showing only less than 1.0 dB compared to the results of experiments using the 31-band analog equalizer. This indicates that the degree of non-flatness of the magnitude response of the headphones used in the experiments, which is also shown Fig. 12, does not result in any noticeable deviations from the expected noise reduction performance computed with the analytically derived equation.

This research was supported by the Chung-Ang University research grant in 2013 and by the Korea National Research Foundation under Grant 20100786. Appendix A. Derivation of the minimum mean squared prediction error The minimum mean-squared prediction error for the second order autoregressive model given in (6) is derived as follows. From (3), the minimum mean-squared prediction error is expressed using (4) as

 2  h i ^  E jemin ðnÞj2 ¼ r2d  E d o ðnjdnD Þ  2  h i ^  2 ¼ E jdðnÞj  E d o ðnjdnD Þ ¼

k¼0

k¼0

A closed-form noise reduction performance equation of the FBANC earmuffs system was derived in this paper. The derivation of the equation took advantage of the simplified and equivalent linear prediction model of the FBANC and its detailed performance

1 X 2 h ðkÞ  r2w ¼ k¼D

¼

"

r2w 2

2 sin xo 1  l

2



cos 2x0  l 2

1  2l cos 2x0 þ l

r2w ð1 þ l2 Þ 2

#

2

1

2

4

ð1  l Þð1  2l cos 2x0 þ l Þ

4

:

ðA:2Þ

The second term is obtained using (5) and (A.2) as

"

2 r2w l2D 1 cos 2x0 Dðcos 2x0  l Þ  sinð2x0 Þ sinð2x0 DÞ  2 2 2 4 2 sin xo 1  l 1  2 cos 2x0  l þ l

(

¼

ðA:1Þ

k¼D

P where dðnÞ ¼ 1 k¼0 hðkÞiðn  kÞ. The first term is obtained by substituting (5) with 1 X 2 h ðkÞ  r2w ¼

5. Conclusions

1 1 X X 2 2 h ðkÞ  r2w  h ðkÞ  r2w ;

#

)

2 r2w l2D 1 cos 2x0  l r2w l2D ð1  cos 2x0 DÞðcos 2x0  l2 Þ þ sinð2x0 Þ sinð2x0 DÞ   þ 2 2 2 4 2 2 4 2 sin xo 1  l 2 sin x0 1  2l cos 2x0 þ l 1  2 cos 2x0  l þ l

"

¼r

2 2D dl

þl "

2D

¼ r2d l

#

r2w ð1 þ l2 Þ

2D 2

2

4

ð1  l Þð1  2l cos 2x0 þ l Þ

2

2



2 sin ðx0 DÞðcos 2x0  l Þ þ sinð2xo Þ sinð2x0 DÞ 2

2

2

2 sin x0  ð1 þ l Þ=ð1  l Þ # 2 2 2 sin ðx0 DÞðcos 2x0  l Þ þ sinð2xo Þ sinð2x0 DÞ : 1þ 2 2 2 2 sin x0  ð1 þ l Þ=ð1  l Þ

quantification using the AR model based parameterization of the noise bandwidth. Two key performance-determining physical

ðA:3Þ

By plugging (A.3) into (A.1), the minimum mean-squared D-step prediction error of dðnÞ is obtained as

60

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h i 2D E jemin ðnÞj2 ¼ r2d  r2d l " # 2 2 sinð2xo Þ sinð2x0 DÞ þ 2sin ðx0 DÞðcos 2x0  l Þ : ðA:4Þ  1þ 2 2 2 2sin x0  ð1 þ l Þ=ð1  l Þ

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